What does Ethan Siegel mean by "The fact that there are no stationary, oscillating in-phase electric and magnetic fields led to Special Relativity."

In a traveling wave, the magnitude of the magnetic field is proportional to the magnitude of the electric field at each point in space (they can't be equal because they have different units, but they are related by $|E| = c|B|$). This means the nodes and antinodes of the two fields are at the same places, and hence the magnetic and electric fields are "in-phase." See this picture.

However, in a standing wave (like between two mirrors in a cavity), the nodes of magnetic field are at the antinodes of the electric field and vice-versa, so they are "out-of-phase." See this picture.

My guess is that the author is saying that there is no standing wave where the nodes and antinodes of the two fields are aligned. That is, there is no way to make a traveling wave stationary, even relative to you when you try to catch up to it.

This relates to Einstein's thought experiment that PM 2Ring linked in the comments.